# Ari Laptev's 70th birthday conference: Recording of talks

Recordings of the virtual conference on the occasion of the 70th birthday of Ari Laptev - August 12 - 13, 2020

Recording of the **greetings to Ari Laptev**

Video-Recording of the talks of:

R. Benguria (Santiago) | A variational formulation for Dirac Operators in bounded domains and applications to spectral geometric inequalities |

M. Esteban (Paris) | Magnetic Interpolation Inequalities in Dimensions 2 and 3 |

R. Frank (Caltech/LMU Munich) | Minimal magnetic fields supporting a zero mode |

E.H. Lieb (Princeton) | Sharpened $L^p$ Triangle Inequalities |

B. Simon (Caltech) | The Tale of a Wrong Conjecture: Borg’s Theorem for Periodic Jacobi Matrices on Trees |

S.T. Yau (Harvard) | Stability and Partial Differential Equations in Mirror Symmetry |

R. Benguria (Santiago)A variational formulation for Dirac Operators in bounded domains and applications to spectral geometric inequalitiesAbstract: In this talk I will present spectral features of the Dirac operator with infi nite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. A non-linear variational formulation to characterize its principal eigenvalue will be presented. This characterization allows for a simple proof of a Szeg \H{o} type inequality as well as a new formulation of a Faber-Krahn type inequality for this operator. Moreover, strong numerical evidence supporting the existence of a Faber-Krahn type inequality, will be given. This talk is based on joint work with Pedro Antunes, Vladimir Lotoreichik, and Thomas Ourmières-Bonafos. |

M. Esteban (Paris)Magnetic Interpolation Inequalities in Dimensions 2 and 3.Abstract: In this talk I will present various results concerning interpolation inequalities, best constants and information about the extremal functions involving Schrödinger magnetic operators in dimensions 2 and 3. The particular, and physical interesting, cases of constant and of Aharonov-Bohm magnetic fields will be discussed in detail. These works have been made in collaboration with D. Bonheure, J. Dolbeault, A. Laptev, and M. Loss. |

R. Frank (Munich)Minimal magnetic fields supporting a zero modeAbstract: We give a lower bound on the L^{3/2} norm of a magnetic field such that the associated Pauli operator has a zero mode. Our bound is within a factor of 2 of the optimal bound. We also discuss two related conformally invariant Sobolev inequalities, one for spinor and one for vector fields, and prove the existence of optimizers. The talk is based on joint work with Michael Loss. |

E.H. Lieb (Princeton)Abstract: Consider the Lp triangle inequality for the norms of two functions, |f+g| \leq |f|+|g|, which is saturated when f=g, but which is poor when f and g have disjoint support. In 2009 Carbery proposed a slightly more complicated inequality for p\geq 2 to take into account the possible orthogonality, or lack of it, of f and g. With Eric Carlen, Rupert Frank and Paata Ivanisvili it has now been proved. In fact, a much stronger version has been proved -- and, moreover, for all -\infty < p < \infty.Sharpened $L^p$ Triangle Inequalities In a recent, subsequent paper with Carlen and Frank an extension to more than two functions was given. |

B. Simon (Caltech)The Tale of a Wrong Conjecture: Borg’s Theorem for Periodic Jacobi Matrices on TreesAbstract: I will begin by reviewing work on removal of eigenvalue degeneracy and its relevance to gap splitting. I’ll next discuss Borg’s theorem. I’ll then describe a framework for discussing periodic Jacobi matrices on trees and possible versions of Borg’s theorem and a recent note that there are counterexamples. Finally, I’ll discuss possible modified conjectures. This includes joint work with Nir Avni and Jonathan Breuer. |

S.T. Yau (Harvard)Stability and Partial Differential Equations in Mirror SymmetryAbstract: This is joint work with Tristan Collins where we studied the concept of Deformed Hermitian Yang Mills equations. It also contains some work that we did with Jacob Adams. The equations arise in the study of mirror symmetry through the SYZ program. The solutions of such equations lead to the concept of stability in algebraic geometry, which seems to be very much related to the concept of Bridgeland stability and the Chern number inequality. |